Problem: Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{p^2 - 7p - 8}{p^2 + p - 72}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{p^2 - 7p - 8}{p^2 + p - 72} = \dfrac{(p + 1)(p - 8)}{(p + 9)(p - 8)} $ Notice that the term $(p - 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p - 8)$ gives: $q = \dfrac{p + 1}{p + 9}$ Since we divided by $(p - 8)$, $p \neq 8$. $q = \dfrac{p + 1}{p + 9}; \space p \neq 8$